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The logarithm,  $   \mathop{\rm ln}  M( \alpha , \beta ) $, 
of the least upper bound of the modulus  $  M( \alpha , \beta ) $
of the bilinear form

$$ 
\sum_{i=1} ^ { m }  \sum_{j=1} ^ { n }  a _ {ij} x _ {i} y _ {j}  $$

on the set

$$ 
\sum_{i=1} ^ { m }  | x _ {i} | ^ {1/ \alpha }  \leq   1,\ \ 
\sum_{j=1} ^ { m }  | y _ {j} | ^ {1/ \beta }  \leq   1
$$

(if  $  \alpha = 0 $
or  $  \beta = 0 $, 
then, respectively,  $  | x _ {i} | \leq  1 $, 
$  i = 1 \dots m $
or  $  | y _ {j} | \leq  1 $, 
$  j = 1 \dots n $) 
is a [[Convex function (of a real variable)|convex function (of a real variable)]] of the parameters  $  \alpha $
and  $  \beta $
in the domain  $  \alpha \geq  0 $, 
$  \beta \geq  0 $
if the form is real  $  ( a _ {ij} , x _ {i} , y _ {j} \in \mathbf R _ {+} ) $, 
and it is a convex function (of a real variable) in the domain  $  0 \leq  \alpha , \beta \leq  1 $, 
$  \alpha + \beta \geq  1 $
if the form is complex  $  ( a _ {ij} , x _ {i} , y _ {j} \in \mathbf C ) $. 
This theorem was proved by M. Riesz [[#References|[1]]].

A generalization of this theorem to linear operators is (see [[#References|[3]]]): Let  $  L _ {p} $, 
$  1 \leq  p \leq  \infty $, 
be the set of all complex-valued functions on some measure space that are summable to the  $  p $-
th power for  $  1 \leq  p < \infty $
and that are essentially bounded for  $  p = \infty $. 
Let, further,  $  T:  L _ {p _ {i}  } \rightarrow L _ {q _ {i}  } $, 
$  1 \leq  p _ {i} , q _ {j} \leq  \infty $, 
$  i = 0, 1 $, 
be a continuous linear operator. Then  $  T $
is a continuous operator from  $  L _ {p _ {t}  } $
to  $  L _ {q _ {t}  } $, 
where

$$ 

\frac{1}{p _ {t} }
  =  1- 
\frac{t}{p _ {0} }
 + 
\frac{t}{p _ {1} }
 ,\ \ 

\frac{1}{q _ {t} }
  =  1- 
\frac{t}{q _ {0} }
 + 
\frac{t}{q _ {1} }
 ,\ \ 
t \in [ 0, 1],
$$

and where the norm  $  k _ {t} $
of  $  T $(
as an operator from  $  L _ {p _ {t}  } $
to  $  L _ {q _ {t}  } $) 
satisfies the inequality  $  k _ {t} \leq  k _ {0}  ^ {1-} t k _ {1}  ^ {t} $(
i.e. it is a logarithmically convex function). This theorem is called the Riesz–Thorin interpolation theorem, and sometimes also the Riesz convexity theorem [[#References|[4]]].

The Riesz convexity theorem is at the origin of a whole trend of analysis in which one studies interpolation properties of linear operators. Among the first generalizations of the Riesz convexity theorem is the Marcinkiewicz interpolation theorem [[#References|[5]]], which ensures for  $  1 \leq  p _ {i} \leq  q _ {i} \leq  \infty $, 
$  i = 0, 1 $, 
the continuity of the operator  $  T:  L _ {p _ {t}  } \rightarrow L _ {q _ {t}  } $, 
$  t \in ( 0, 1) $, 
under weaker assumptions than those of the Riesz–Thorin theorem. See also [[Interpolation of operators]].

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  M. Riesz,   "Sur les maxima des formes bilinéaires et sur les fonctionnelles linéaires"  ''Acta Math.'' , '''49'''  (1926)  pp. 465–497</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  G.H. Hardy,   J.E. Littlewood,   G. Pólya,   "Inequalities" , Cambridge Univ. Press  (1934)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  G.O. Thorin,   "An extension of a convexity theorem due to M. Riesz"  ''K. Fysiogr. Saallskap. i Lund Forh.'' , '''8''' :  14  (1936)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  E.M. Stein,   G. Weiss,   "Introduction to Fourier analysis on Euclidean spaces" , Princeton Univ. Press  (1971)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  J. Marcinkiewicz,   "Sur l'interpolation d'opérateurs"  ''C.R. Acad. Sci. Paris'' , '''208'''  (1939)  pp. 1272–1273</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  S.K. Krein,   "Interpolation of linear operators" , Amer. Math. Soc.  (1982)  (Translated from Russian)</TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top">  H. Triebel,   "Interpolation theory" , Springer  (1978)</TD></TR></table>